Modulus and strength of filled and crystalline polymers

Polymer reports References 1 Sperling,L. H. and Friedman, D. W. J. Polym. Sci., Part A 1969, 7, 425

2

3 4 5 6 7 8 9 10

Grates,J. A., Thomas, D. A., Hiekey, E. C. and Sperling, L. H.

J. AppL Polym. $ci. 1975, 19, 1731 Akovali,G., Biliyar, K. and Shen, M. J. Appl. Polym. Sci. 1976, 20, 2419 Kim, S. C., Klempner, D., Frisch, K. C., Radigan, W. and Frisch, H. L. Macromolecules 1976, 9, 258 Touhsaent, R. E., Thomas, D. A. and Sperling, L. H. J. Polym. Sci. (C} 1974,46, 175 Manson,J. A. and Sperling, L. H. 'Polymer Blends and Composites' Plenum Press, New York, 1976 Klempner, D. and Frisch, K. C. 'Polymer Alloys' Plenum Press, New York, 1977 Paul,D. R. and Newman, S. 'Polymer Blends' Academic Press, New York, 1978, Vol. 2, p 1 Kim,S. C., Klempner, D., Frisch, K. C., Frisch, H. L. and Ghiradella, H. Polym. Eng. ScL 1975, 15,339 Donateili,A. A., Thomas, D. A. and Sperling, L. H. 'Recent Advances in Polymer Blends, Grafts and Blocks' Plenum Press, New York, 1974

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Huelk,V., Thomas, D. A. and Sperling, L. H. Macromolecules 1972, 5,340 Hourston, D. J. and Hughes, I. D.J. AppL Polym. Sei. 1977, 21, 3099 Williams,M. L. and Ferry, J. D. J. Colloid ScL 1955, 10, 474 Thurn, H. and Wolf, K. Kolloid-Z. 1958, 156, 21 Heijboer,J. 'Physics of Non-Crystalline Solids' North Holland, Amsterdam, 1965, p 231 Read, B. E.Polymer 1964, 5, 1 Mead,D.J. andFuoss, R.M.J. Am. Chem. Soc. 1942,64, 2389 Mikhailov,G. P. Zh. Tekh. Fiz. 1951,21, 1395 Scheiber, D. J. Dissertation, University of Notre Dame, Indiana, 1957 Brouckere, L. and Offergeld, G. J. Polym. ScL 1958, 30, 105 Ishida,Y. Kolloid-Z. 1961, 174, 124 Williams,G. Trans. Faraday Soc. 1964, 60, 1548, 1556 McCrum,N. G., Read, B. E. and Williams, G. 'Anelastic and Dielectric Effects in Polymeric Solids' Wiley, London, 1967 Schmieder, K. and Wolf, K. Kolloid-z 1953, 134, 149 Thurn, H. and Wolf, K. Kolloid.Z. 1956, 148, 66

Modulus and strength of filled and crystalline polymers* T. S. C h o w

Xerox Corporation, J. C. Wilson Center for TechnologF, Rochester, N. Y. 14644, USA (Received 22 June 1979)

Introduction When inclusions have a higher elastic modulus than the matrix, the chief effect of the filler is to increase the moduli of the composite. Theories of varying degrees of ref'mement have been reviewed 1,2 to account quantitatively for spherical filler particles and uniaxially oriented infinitely long fibres. For aligned chopped fibre composites, Halpin and Tsai 2 introduced a simple set of approximate equations which were later employed by Halpin and Kardos 3 to study the elastic moduli of partially crystalline polymers. Recently, Porter and his colleagues 4,s have found that the Halpin-Tsai equation does not provide a correct estimation of the crystal shape in semicrystalline polyethylene. Based on a recent model developed by the author 6'7, the relation of the modulus of semicrystalline polymers and the shape of crystallites will be discussed here. Because stress concentration factors are very critical parameters in the study of fracture and strength of heterogeneous materials, a relation is proposed for their determination which includes the effects of particle shape and volume concentration of tillers.

tensile modulus

Modulus The modulus of a crystalline polymer has been considered by Takayanagi s. His series-parallel model gives the effective

where ~ = (Ef/Em - 1)/(Ef/Ern + e), e = 2(l/d) and ¢ is the volume fraction of crystals. From electron diffraction measurements of ultradrawn polyethylene, Crystal and Southern 9 observed extended chain crystals approximately 2 0 0 - 2 5 0 A in diameter and 5000 A long. The properties of high-density polyethylene are s

* This paper was a part of t.he presentation at the 2nd Joint Meeting of U.S. and Japan Societies of Rheology, Symposium on Suspensions and Filled Systems, Kona, Hawaii, April 1979 0032-3861/79/121576-03502.00 © 1979 IPC BusinessPress 1576 POLYMER, 1979, Vol 20, December

~ EH =

0Era +(1 - O ) E f

1-~]

(1)

+ ~ E l"

where the subscripts f and m identify the crystalline and amorphouse phases. The amorphous region is of volume ~0 and the crystalline region (1 - ~0). The basic problem with the model is how to decide the values of tp and 0. In an alternative approach, Halpin and Kardos 3 employ the Halpin-Tsai equation 2 to determine the modulus of semicrystalline polymers from the crystal shape characterized by the aspect ratio lid. For a uniaxially oriented morphology the tensile modulus is given by

Eli/Era = (1 + er/40/(1 - r#~)

(2)

Polymer reports El= 2.4 × 1012 dyn cm -2 E m = l x 1 0 9 d y n c m -2

(3)

¢=0.85 Mead, Porter, Southern and Weeks 4's found the crystal shape, calculated from equation (2), much greater than the measured aspect ratio lid = 2 0 - 2 5 . This conclusion is not entirely surprising because equation (2) was reduced from Hermans' solution 2 to a simple approximate form containing a geometric parameter e. The parameter is determined for ordinary short-fibre reinforced plastics with the ratio of Ef/Em rarely exceeding 1000. The ratio of moduli for semicrystalline polyethylene is 2400. Thus, the Halpin-Tsai equation originally proposed for composite materials may not be adequate for semicrystalline polymers. A recent model by the author 7 has considered fully the effects of particle shape and concentration of two phase anisotropic heterogeneous materials. There is no fitting parameter in the model. Both the filler and matrix are assumed isotropic. Fillers are treated as identical spheroids

(x 2 + x~)/a 2 + x2/c 2= 1

not include the concentration dependence. The analysis, to include the filler-ffiler interaction, follows exactly the mean field theory developed for elastic moduli 7 (see equation (5)) and thermal expansion 14. The spheroids are assumed to be firmly bonded to the matrix and the composite as a whole is macroscopically homogeneous. The volume average of internal stresses, (Opq), inside prolate spheroids due to uniform applied stress o.4 q can be determined from 7 ( I - qS)[(~f - ~rn ) Skkpq ~)i/+ 2(/.tf - Idrn)Sijpq ] (Opq }

+ )~m(O}Si/+ 21arn(Oil) = ~,foASij + 212f~i1

where X = k - 2/1/3 is the Lame constant, 8i! is the Kronecker delta, Sijpq is Eshelby's tensor 7' ~s and e is the dilatation. There are five independent equations with (023) = (o13>. Consider a uniform tensile stress oA3 as the only nontrivial stress applied to the system. The stress concentration factor is

(4)

S-

(o33) "

-

(kf/km)G1 + 2(#f/#m)K1

o3A3 When the corresponding axes are aligned and the spatial distribution of the fiUers is random and homogeneous, the medium as a whole becomes transversely isotropic about the x 3 axis. The effective longitudinal Young's modulus Ellis given by Eli

-

I + (kf/krn - 1)G1 + 2(#Ilium - 1)K1

Em

2K1G3 + G1K3

-

1)(1 --q~)ai

where K i and Gi (i = 1,3) are given in equation (6). The above equation contains both shape and concentration effects of fillers. When 4~-+ 0 and p = 1, equation (8) becomes

(i 1,3)

El= 73.1 x 1010 dyn cm -2, /af= 0.22

=

(i

=

(9)

which is the classical Goodier Equation n. For illustration, consider glass-filled poly(phenylene oxide) (PPO) possessing the following properties:

(6) Gi = 1 + (Ul/Urn - 1)(1 - ~ ) &

(8)

2 K 1 G 3 +G1K3

S = [(kf/k m )/K + 2(IJf[#m )/C ]/3 (5)

where k is the bulk modulus,/~ is the shear modulus, K i = 1 + (kf/km

(7)

(10)

1,3)

ai and j3i are functions of the aspect ratio p = c/a and Poisson's ratio Vm of the matrix (see Appendix). Equation (5) compares well with experiments for reinforced plastics 7. For spherical fillers (p = 1), K 1 = K 3 = K, G 1 = G 3 = G and equation (5) reduces to the well known Kerner equation 7. When the aspect ratio p + oo equation (5) becomes the limiting upper bound for semicrystalline polymers 3. Mead and Porter 4 measured the tensile modulus of ultraoriented high-density polyethylene: Ejl = 50 × 1010 dyn cm -2. Using equation (3) and assuming uf = 0.25 and Urn = 0.45, the calculated shape of crystallites from equation (5) is p = 30 which is very close to the measured aspect ratio (lid = 20 - 25). This suggests the potential use of equation (5) in the structure-property estimation of a semicrystalline polymer.

Stress concentration The ratio of the maximum stress to the applied stress defines the stress concentration factor. Consider aligned prolate spheroids (see equation (4) with e > a) in a matrix of lower modulus under a uniform tensile stress parallel to the x3 axis. The maximum stress occurs at (Xl, x2, x3) = (0,0,c) where cohesive and/or adhesive failure takes place. Therefore, the determination of stress concentration factor is essential for heterogeneous materials. Traditionally, the stress concentration calculation has been limited to a single isolated inclusion n-*3 which does

Ern = 2.6 x 1010 dyn cm -2, /2rn = 0.35 The stress concentration factor is calculated from equation (8) as a function of p and 4. The general behaviour is shown in Figure 1. It reveals that the concentration dependence diminishes for large values of the aspect ratio. 50

~D 10 P

u

I

lOO

lO Aspect ratio, p=c/o

Figure I

Effects of p and q) on the stress concentration factor.

A, q)--* 0; B, q~ = 0.4

P O L Y M E R , 1979, Vol 20, December

1577

Polymerreports References

1

Manson,J. A. and Sperling, L. H. 'Polymer Blends and Composites', Plenum Press, N.Y., 1976, Ch. 12 Ashton, J. E., Halpin, J. C. and Petit, P. H. 'Primer on Composite Materials: Analysis', Technomic, Stamford, Conn., 1969, p 77 Halpin,J. C. and Kardos, J. L. Z AppL Phys. 1972, 43, 2235 Mead,W. T. and Porter, R. S. J. AppL Phys. 1976, 47, 4278 Porter, R. S., Southern, J. H. and Weeks, N. Polym. Eng. Sei. 1975, 15,213 Chow, T. S. J. Appl. Phys. 1977, 48, 4072 Chow, T. S. J. Polym. Sci. (Polym. Phys. Edn.) 1978, 16, 959 Takayanagi, M., Harima, H. and lwata, Y. Rep. Prog. Polym. Phys. Jpn 1963, 6,121 Crystal, R. C. and Southern, J. H. J. Polym. Sci. A2, 1971, 9, 1641 Hermans,J. J. Proc. R. Acad. Amsterdam 1967, B70, 1 Goodier, J. N.J. Appl. Mech. 1933, 1, 39 Edwards, R. H.J. Appl. Mech. 1951,18,19 Argon, A. S. Fibre ScL Technol. 1976, 9, 265 Chow, T. S. Jr. Polym. Sci. (Polym. Phys. Edn.) 1978, 16, 967 Eshelby, J. D.Proc. R. Soc. London Ser. A. 1957, 241, 376

2 3 4 5 6 7 8 9 10 11 12 13 14 15

4n - 3 1 ) i~ ~ Q-4(1-

~1 = 47r/3

2~r)g

133 = [47r/3 - (4n - 31)p2[(1 - 02)] Q + (41r - I ) R where

Q-

3

1

--,R-

81r l - v

m

1 1-2v m 87r l - v

m

and 2~rp (1 - p2)3/2

[ c o s - l p -- p(1 -- p2)1/2]

f o r p < 1,

[/9(/92 -- 1)1/2 _ c o s h - l p ]

forp>

r= 2rrp ( p 2 _ 1)3/2

1

When p ~ 1, we have

Appendix The parameters for equation (6) are listed in the following:

% ( 1 + Vml al = a 3 = a = 3~1 - - V m ]

otI = 41rQ/3 - 2(21r - I ) R

fll = 133 = ~ =

a 3 = 4~Q/3 + 4(1 - rr)R

/

15~_-i-S~m 1 ]

ERRATUM 'Proton spin-lattice relaxation in vinyl polymers and an application of a solvable model of polymer dynamics' F. Heatley* and J. T. Bendlert There is an error in the definition of the relaxation coefficients in equation (3) of reference 1, concerned with coupled proton spin-lattice relaxation in vinyl polymers. The equation defining the cross-relaxation coefficients TAX and TXA is in error by a factor of 2, and should read 1

1 B

_

_

TAx 2TxA

12J(coA + coX) - 2J(coA -- coX) --

K

RIX

This error is solely a transcription error. Relaxation parameters reported in reference 1 were evaluated using the correct form. Unfortunately, the incorrect expression was employed in a subsequer.t application of a jump model of polymer motion 2. As a consequence of the correction above, calculated nuclear Overhauser enhancements given in Table 111 of reference 2 should be multiplied by 2. The agreement with experiment is much improved. (Note also that the entry for the parameter N A at 10°C in the fourteenth column of Table 111 has been incorrectly printed as - 0 . 9 instead of -0.09.) References

1 Heatley, F. and Cox, M. K. Polymer 1977, 18, 225 2 Bendler, J. T. and Yaris, R. Macromolecules 1978, 11,650

* Department of Chemistry, University of Manchester, Manchester, M13 9PL, UK t General Electric, R and D Center, Schenectady, N. Y. 12301, USA

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POLYMER, 1979, Vol 20, December